Fault monitoring method for sewage treatment process based on fuzzy width adaptive learning model

ABSTRACT

The invention discloses a sewage treatment process fault monitoring method based on fuzzy width adaptive learning model. Including “offline modeling” and “online monitoring” two stages. “Offline modeling” first uses a batch of normal data and 4 batches of fault data as training samples to train the network offline and label the data. After the network training is completed, the weight parameters are obtained for online monitoring. “Online monitoring” includes: using newly collected data as test data, using the same steps as offline training networks for online monitoring. The output result of online monitoring adopts one-hot encoding to realize zero-one discrimination of the output result of online monitoring, so as to realize fault monitoring. The present invention only needs to increase the number of enhanced nodes, reconstruct in an incremental manner, and does not need to retrain the entire network from the beginning. It can complete the network training in a short time and realize the rapid fault monitoring in time, which has high practical application value.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of International Application No.PCT/CN2019/124479, filed on Dec. 11, 2019, which claims priority toChinese Patent Application No. 201911225929.3, filed on Dec. 3, 2019.The contents of the above applications are hereby incorporated byreference in their entireties and form a part of this specification.

TECHNICAL FIELD

The present invention relates to the technical field of fault monitoringbased on data-driven, in particular to a fault monitoring technology forsewage treatment processes. The fault monitoring method of the presentinvention for the intermittent process is a specific application in thefault monitoring of the sewage treatment process.

BACKGROUND ART

In recent years, with the advancement of science and technology, theprocess of industrialization and urbanization has been accelerating, thenumber of people has continued to increase, and the pollution of theecological environment has become increasingly serious. Among them, asan essential water resource for human life, due to its shortage andserious pollution, the protection of the water environment has becomethe consensus of today's society. People are seeking a practical way toprotect the water environment, and sewage treatment plants are used aswater resources. The key link in the environmental protection chain hasreceived extensive attention from the outside world. The primary purposeof a sewage treatment plant is to ensure that its final product, thatis, the quality of the effluent, can meet the prescribed dischargestandards, and then improve the efficiency of sewage treatment on thisbasis to save costs. The prerequisite for achieving this goal is thatthe sewage treatment plant must be in a safe and stable operating state.Therefore, it is necessary to establish an effective process monitoringprogram to detect abnormal phenomena in time.

At present, neural network-based fault monitoring methods based onmachine learning are widely used in the field of fault monitoring insewage treatment processes, such as DBN (Deep Belief Network), DBM (DeepBoltzmann Machine), ANN (Artificial Neural Network), AE (Auto-encoder))and other methods. The currently popular deep neural networks have verypowerful performance, but due to the large number of hyperparameters andcomplex structures involved, the training of the network is a verytime-consuming process. In addition, the complexity of the deep networkstructure itself makes theoretical analysis of its deep structure verydifficult. At present, most researches focus on adjusting parameterssuch as network weights or stacking more layers to obtain higheraccuracy. Therefore, it takes a lot of time and expensive equipment andother hardware resources. Modern fast-paced industrial productionprocess is difficult to meet these conditions.

SUMMARY

In order to solve the above problems, a fault monitoring method forsewage treatment process based on fuzzy width adaptive learning model isproposed. The method includes a set of first-order TS fuzzy subsystemand enhanced node layer. First, each fuzzy subsystem processes the inputdata, and then connects the processed intermediate values as vectors andsends them to the enhanced node group for nonlinear conversion.Secondly, the output of the enhancement layer and the defuzzificationoutput of the fuzzy subsystem are used to generate the final modeloutput. The parameters of the model include the enhanced node group andthe final output connection weight. The coefficients of the subsequentparts of fuzzy rules in each fuzzy subsystem can be quickly calculatedby pseudo-inverse calculations. Different from the traditional deepnetwork, this method only needs to reconstruct the model in anincremental manner, without retraining entire network from scratch. Onthe premise of ensuring that the accuracy meets the requirements, thecharacteristic of fuzzy width learning system to continuously adjustitself with the production process makes it possible to realize onlinefault monitoring sewage treatment process.

The technical scheme and implementation steps adopted by the presentinvention include “network training phase” and “online monitoringphase”. The specific steps are as follows:

A. Network Training Stage:

1) Collect historical data {X, Y} of the sewage treatment process astraining data, where X=(x₁, x₂, . . . , x_(N))^(T)∈

^(N×M), Y=(y₁, y₂, . . . , y_(C))^(T)∈

^(N×C), Y is used as label data, the label is 0 during the normal periodof the process, and the label data is 1 during the fault period. N isthe number of input samples, M is the number of variables, and C is thedimension of the corresponding output (in this method, the outputdimension is 2, which means normal and faulty states).

2) For input data X=(x₁, x₂, . . . , x_(N))^(T)∈

^(N×M), X_(s)=(x_(s1), xs2, . . . , x_(sM)), s=1, 2, . . . , N. Themodel has n fuzzy subsystems, m enhanced node groups, and the i-th fuzzysubsystem contains K fuzzy rules. The first-order TS fuzzy model isused, where the i-th fuzzy subsystem is taken as an example (i=1, 2, . .. , n). When the input data is the s-th training sample, the output ofi-th fuzzy subsystem and k-th fuzzy rule is:

$z_{sk}^{i} = {{f_{k}^{i}\left( {x_{s\; 1},x_{s\; 2},\ldots\mspace{14mu},x_{sM}} \right)} = {\sum\limits_{t = 1}^{M}{\alpha_{kt}^{i}x_{st}}}}$

Wherein f_(k) ^(i) is a polynomial of the first degree, α_(kt) ^(i) is aparameter obtained by initialization which obeys uniform distribution[0,1], t=1, 2, . . . , M, k=1, 2, . . . , K is the number of variables.

3) For the s-th training sample, calculate weighted strength ω_(sk) ^(i)of k-th fuzzy rule of the i-th fuzzy subsystem:

$\omega_{sk}^{i} = \frac{\tau_{sk}^{i}}{\sum\limits_{k = 1}^{k_{i}}\tau_{sk}^{i}}$

Where τ_(sk) ^(i) represents weighted strength of the kth fuzzy rule ofi-th fuzzy subsystem (k=1, 2, . . . , K) when the input training sampleis Xs (xs=(xs1, xs2, . . . , xsM)), the formula is as follows:

$\tau_{sk}^{i} = {\prod\limits_{t = 1}^{M}{\mu_{kt}^{i}\left( x_{st} \right)}}$

Where μ_(kt) ^(i)(x_(st)) is the Gaussian membership function (t=1, 2, .. . , M) corresponding to the input sample X_(st), and its formula isexpressed as:

${\mu_{kt}^{i}\left( x_{st} \right)} = e^{- {(\frac{x_{st} - c_{kt}^{i}}{\delta_{kt}^{i}})}^{2}}$

Wherein c_(kt) ^(i) and δ_(kt) ^(i) are parameters, and the value ofc_(kt) ^(i) is the value of cluster center obtained by C-meansclustering, δ_(kt) ^(i)=1

4) Calculate the output Z^(n) of n fuzzy subsystems corresponding to sthtraining sample:Z ^(n)=(Z ₁ ,Z ₂ , . . . ,Z _(n))∈

^(N×(k) ¹ ^(+k) ² ^(+ . . . k) ^(n) ⁾

Wherein, Z₁, Z₂, . . . , Z_(n) respectively correspond to the output ofn subsystems, k₁, k₂, . . . , k_(n) respectively correspond to thenumber of fuzzy rules of n fuzzy subsystems, and the sth trainingsample.

Taking i-th fuzzy subsystem as an example, for all training samples, theoutput matrix Z_(i) is obtained by the following formula:Z _(i)=(Z _(1i) ,Z _(2i) , . . . ,Z _(Ni))^(T)∈

^(N×K) ,i=1,2, . . . ,n

Wherein, Z_(1i), Z_(2i), . . . , Z_(Ni) respectively represent theoutput vector of i-th fuzzy subsystem for N training samples. Takings-th sample as an example, the output vector of i-th fuzzy subsystem canbe obtained by the following formula:Z _(si)=(ω_(s1) ^(i) z _(s1) ^(i),ω_(s2) ^(i) z _(s2) ^(i), . . .,ω_(sk) ₁ ^(i) z _(sk) ₁ ^(i))

Wherein, z_(sk) ^(i) and ω_(sk) ^(i) are obtained by 2) and 3)respectively, and k=1, 2, . . . , K.

5) Input the output Z^(n) obtained in 4) into m enhanced node groupsmentioned in 2) for nonlinear conversion. Wherein, the j-th enhancednode group has Lj neurons (j=1, 2, . . . , m). For m enhanced nodegroups, the output matrix can be expressed as:H ^(m)=(H ₁ ,H ₂ , . . . ,H _(m))∈

^(N×(L) ¹ ^(+L) ² ^(+ . . . L) ^(m) ⁾

Wherein, L₁, L₂, . . . , L_(m) respectively represent the number ofneurons contained in m enhanced node groups, H₁=ξ_(j)(Z^(n)W_(j)+β_(j))∈

^(N×L) ^(j) is the output matrix of j-th enhanced node group, and Wjrepresents connection weight between fuzzy subsystem output and thecorresponding enhanced node group, βj represents the deviation betweenoutput of fuzzy subsystem and j-th enhanced node group, the values of Wjand βj are randomly generated from [0,1], ξj is the activation function,and the formula is ξj=tan h(·).

6) Calculate the defuzzification output of fuzzy subsystem. Then fors-th training sample X_(s), the defuzzification output vector of i-thfuzzy subsystem can be expressed as:

$F_{si} = {{\sum\limits_{k = 1}^{k_{i}}{\omega_{sk}^{i}z_{sk}^{i}}} = {\sum\limits_{k = 1}^{k_{i}}\left( {\omega_{sk}^{i}{\sum\limits_{t = 1}^{M}{\alpha_{kt}^{i}x_{st}}}} \right)}}$

Introduce parameter δ_(kc) ^(i) to the result part of each fuzzy rule.The parameter value is obtained by pseudo-inverse operation method, andC represents dimension of output. Since Y=∈

^(N×C) has C dimensions, each fuzzy subsystem is a multi-output model.After the parameter is introduced, its new output is:

$\begin{matrix}{F_{si} = \left( {{\sum\limits_{k = 1}^{k_{i}}{\omega_{sk}^{i}\left( {\sum\limits_{t = 1}^{M}{\delta_{k1}^{i}\alpha_{kt}^{i}x_{st}}} \right)}},\ldots\mspace{14mu},{\sum\limits_{k = 1}^{K}{\omega_{sk}^{i}\left( {\sum\limits_{t = 1}^{M}{\delta_{kC}^{i}\alpha_{kt}^{i}x_{st}}} \right)}}} \right)} \\{= {\sum\limits_{t = 1}^{M}{\alpha_{kt}^{i}{x_{st}\left( {\omega_{s\; 1}^{i},\omega_{s2}^{i},\ldots\mspace{14mu},\ \omega_{sK}^{i}} \right)}\begin{pmatrix}\delta_{11}^{i} & \ldots & \delta_{1C}^{i} \\\vdots & \ddots & \vdots \\\delta_{K1}^{i} & \ldots & \delta_{KC}^{i}\end{pmatrix}}}}\end{matrix}$

Where s=1, 2, . . . , N, then, for all training samples X, the output ofthe i-th fuzzy subsystem is:

$F_{i} = \ {\left( {F_{1i},F_{2i},\ldots\mspace{14mu},F_{Ni}} \right)^{T}\overset{\Delta}{=}{{D\;\Omega^{i}\delta^{i}} \in {\mathbb{R}}^{N \times C}}}$${{{Where}\mspace{14mu} D} = {{diag}\left\{ {{\sum\limits_{t = 1}^{M}{\alpha_{kt}^{i}x_{1t}}}\ ,\ldots\mspace{14mu},\ {\sum\limits_{t = 1}^{M}{\alpha_{kt}^{i}x_{Nt}}}} \right\}}},{\Omega^{i} = \begin{pmatrix}\omega_{11}^{i} & \ldots & \omega_{1K}^{i} \\\vdots & \ddots & \vdots \\\omega_{N1}^{i} & \ldots & \omega_{NK}^{i}\end{pmatrix}},{\delta^{i} = \begin{pmatrix}\delta_{11}^{i} & \ldots & \delta_{1C}^{i} \\\vdots & \ddots & \vdots \\\delta_{K1}^{i} & \ldots & \delta_{KC}^{i}\end{pmatrix}}$

Thus, the output of n fuzzy subsystems can be obtained:

$F^{n} = {{\sum\limits_{i = 1}^{n}F_{i}} = {{\sum\limits_{i = 1}^{n}{D\Omega^{i}\delta^{i}}} = {{{D\left( {\Omega^{1},\ \Omega^{2},\ \ldots\mspace{14mu},\ \Omega^{n}} \right)}\begin{pmatrix}\delta^{1} \\\vdots \\\delta^{n}\end{pmatrix}} = {{D\Omega\Delta} \in {\mathbb{R}}^{N \times C}}}}}$

Wherein, Ω=(Ω¹, Ω², . . . , Ω^(n))∈

^(N×nk) is the matrix containing the weighted strengthen ω_(sk) ^(i)obtained in 3). Δ=((δ¹)^(T), . . . , (δ^(n))^(T))^(T)∈

^(nK×C)

7) Calculate the final output result. Weight between fuzzy subsystem andoutput is set to 1, and the weight between enhanced node group andoutput is set to W_(e)∈

^((L) ¹ ^(+L) ² ^(+ . . . L) ^(m) ^()×C). Multiply the H^(m) obtained in4) and F^(n) obtained in 5) respectively. The weights are then summed toget the final output result, the formula is as follows:

$\hat{Y} = {{F^{n} + {H^{m}W_{e}}} = {{{D\Omega\Delta} + {H^{m}W_{e}}} = {{\left( {{D\Omega},H^{m}} \right)\begin{pmatrix}\Delta \\W_{e}\end{pmatrix}}\overset{\Delta}{=}{{\Delta\left( {{D\Omega},H^{m}} \right)}W}}}}$

For a given training target Y, the matrix W can be calculated bypseudo-inverse:W=(DΩ,H ^(m))⁺ Y

Wherein, so far, the network model training is complete.

B Online Monitoring:

8) During online monitoring, set the new data X_(new)=(x_(new,1),x_(new,2), . . . , x_(new,N))^(T)∈

^(N×M), X_(new,s)=(X_(new,1), X_(new,2), . . . , X_(new,M)), s=1, 2, . .. , N. For n fuzzy subsystems in the training part, taking i-thsubsystem as an example, using first-order T-S fuzzy system, then:

$z_{{new},{ski}}^{i} = {{f_{k}^{i}\left( {x_{{new},{s\; 1}},x_{{new},{s\; 2}},\ldots\mspace{14mu},x_{{new},{sM}}} \right)} = {\sum\limits_{t = 1}^{M}{\alpha_{kt}^{i}x_{{new},{st}}}}}$

The value of α_(kt) ^(i) is the value corresponding to the trainingpart.

9) For s-th test sample, the weighted strength of i-th fuzzy subsystemis as follows:

$\omega_{{new},{sk}}^{i} = \frac{\tau_{{new},{sk}}^{i}}{\sum\limits_{k = 1}^{K}\;\tau_{{new},{sk}}^{i}}$

Where τ_(new,sk) ^(i) represents the strength of k-th fuzzy rule of i-thfuzzy subsystem, which is obtained by the following formula:

$\tau_{{new},{sk}}^{i} = {\prod\limits_{t = 1}^{M}\;{\mu_{kt}^{i}\left( x_{{new},{st}} \right)}}$

Where μ_(kt) ^(i)(x_(new,st)) is the Gaussian membership functioncorresponding to the input sample X_(new,st) (t=1, 2, . . . , M), andits formula is expressed as:

${\mu_{kt}^{i}\left( x_{{new},{st}} \right)} = e^{- {(\frac{x_{{new},{st}} - c_{kt}^{i}}{\delta_{kt}^{i}})}^{2}}$

Wherein, c_(kt) ^(i) and δ_(kt) ^(i) are parameters, and their valuesare the same as corresponding parameter values of the training part.

10) For all test samples, find the output Z_(new) ^(n) of n fuzzysubsystems. The formula is as follows:Z _(new) ^(n)=(Z _(new1) ,Z _(new2) , . . . ,Z _(newn))∈

^(N×nK)

The output Z_(newi) of i-th fuzzy subsystem can be expressed as follows:Z _(newi)=(Z _(new,1i) ,Z _(new,2i) , . . . ,Z _(new,Ni))^(T)∈

^(N×K) ,i=1,2, . . . ,n

Taking s-th training sample among all N samples as an example, theoutput vector Z_(new,si) of i-th fuzzy subsystem can be expressed as:Z _(new,si)=(ω_(new,s1) ^(i) z _(new,s1) ^(i),ω_(new,s2) ^(i) z_(new,s2) ^(i), . . . ,ω_(new,sK) ^(i) z _(newsK) ^(i))

Where (s=1, 2, . . . , N)

Wherein, z_(new,sk) ^(i) and ω_(new,sk) ^(i) are obtained by 8) and 9)respectively, k=1, 2, . . . , K.

11) The output Z_(new) ^(n) obtained in 10) is subjected to nonlinearconversion.

Then for m enhanced node groups, the output matrix of enhanced layer canbe expressed as:H _(new) ^(m)=(H _(new1) ,H _(new2) , . . . ,H _(newm))∈

^(N×(L) ¹ ^(+L) ² ^(+ . . . L) ^(m) ⁾

Where H_(newj)=ξ_(j)(Z_(new) ^(n)W_(j)+β_(j))∈

^(N×L) ^(j) is the output matrix of j-th enhanced node group (j=1, 2, .. . , m), and the values of Wj and βj correspond to the network trainingpart.

12) Calculate the defuzzification output of fuzzy subsystem.

After introducing parameter δ_(kc) ^(i) to the result part of each fuzzyrule, c=1, 2, . . . , C. For s-th training sample, defuzzificationoutput vector of i-th fuzzy subsystem can be expressed as:

$\begin{matrix}{F_{newsi} = \left( {\sum\limits_{k = 1}^{K_{i}}\;{{\omega_{sk}^{i}\left( {\sum\limits_{t = 1}^{M}\;{\delta_{k\; 1}^{i}\alpha_{kt}^{i}x_{newst}}} \right)}\mspace{14mu}\ldots\mspace{14mu}{\sum\limits_{k = 1}^{K_{i}}\;{\omega_{sk}^{i}\left( {\sum\limits_{t = 1}^{M}\;{\delta_{kC}^{i}\alpha_{kt}^{i}x_{newst}}} \right)}}}} \right)} \\{= {\sum\limits_{t = 1}^{M}\;{\alpha_{kt}^{i}{x_{newst}\left( {\omega_{s\; 1}^{i},\omega_{s\; 2}^{i},\ldots\;,\omega_{sK}^{i}} \right)}\begin{pmatrix}\delta_{11}^{i} & \cdots & \delta_{1C}^{i} \\\vdots & \ddots & \vdots \\\delta_{K\; 1}^{i} & \cdots & \delta_{KC}^{i}\end{pmatrix}}}}\end{matrix}$

Then, for all training samples X, the output of i-th fuzzy subsystem is:

$F_{newi} = {\left( {F_{{new},{1i}},F_{{new},{2i}},\ldots\;,F_{{new},{Ni}}} \right)^{T}\overset{\Delta}{=}{{D_{new}\Omega_{new}^{i}\delta^{i}} \in {\mathbb{R}}^{N \times C}}}$${{{Wherein}\mspace{14mu} D_{new}} = {{diag}\left\{ {{\sum\limits_{t = 1}^{M}\;{\alpha_{kt}^{i}x_{{new},{1t}}}},\ldots\;,{\sum\limits_{t = 1}^{M}\;{\alpha_{kt}^{i}x_{{new},{Nt}}}}} \right\}}},{\Omega_{new}^{i} = \begin{pmatrix}\omega_{{new},11}^{i} & \cdots & \omega_{{{new}\; 1},K}^{i} \\\vdots & \ddots & \vdots \\\omega_{{new},{N\; 1}}^{i} & \cdots & \omega_{{new},{NK}}^{i}\end{pmatrix}},{\delta^{i} = \begin{pmatrix}\delta_{11}^{i} & \cdots & \delta_{1C}^{i} \\\vdots & \ddots & \vdots \\\delta_{K\; 1}^{i} & \cdots & \delta_{KC}^{i}\end{pmatrix}}$

Thus, the output of n fuzzy subsystems can be obtained:

$F_{new}^{n} = {{\sum\limits_{i = 1}^{n}\; F_{newi}} = {{\sum\limits_{i = 1}^{n}\;{D_{new}\Omega_{new}^{i}\delta^{i}}} = {{{D_{new}\left( {\Omega_{new}^{1},\ldots,\Omega_{new}^{n}} \right)}\begin{pmatrix}\delta^{1} \\\vdots \\\delta^{n}\end{pmatrix}} = {{D_{new}\Omega_{new}\Delta} \in {\mathbb{R}}^{N \times C}}}}}$

Wherein, Ω_(new)=(Ω_(new) ¹, . . . , Ω_(new) ^(n))∈

^(N×nK) is the matrix containing weighted strength ω_(new,sk) ^(i)obtained in 9).Δ=((δ¹)^(T), . . . ,(δ^(n))^(T))^(T)∈

^(nK×C)

13) Calculate the final output value. Set the weight between combinedoutput of connection enhancement node to W_(e)∈

^((L) ¹ ^(+L) ² ^(+ . . . L) ^(m) ^()×C) whose value is obtained fromtraining phase, and the weight between fuzzy subsystem and output is 1,multiply H_(new) ^(m) obtained from 11) and F_(new) ^(n) obtained from12) by their respective weights, and then sum them to get final output.

The formula is as follows:Ŷ=F _(new) ^(n) +H _(new) ^(m) W _(e)

(D _(new)Ω_(new) ,H _(new) ^(m))W

The value of W corresponds to the value of parameter matrix of networktraining part.(D _(new)Ω_(new) ,H _(new) ^(m))=((D _(new)Ω_(new) ,H _(new) ^(m))^(T)(D_(new)Ω_(new) ,H _(new) ^(m)))⁻¹(D _(new)Ω_(new) ,H _(new) ^(m))^(T)

14) The output value Ŷ is converted to zero by one-hot encoding method.When it is 0, it means that the operation is in a normal state. When theoutput value is 1, it means that the operation is faulty, so as torealize fault monitoring.

Beneficial Effect

The present invention uses the historical data of sewage treatmentprocess to train the network model. When training the model, normalprocess data is labeled as 0 and the fault period data is labeled as 1,the network parameters obtained by training are used for online faultmonitoring.

Compared with current popular deep network, the present invention onlyneeds to increase the number of enhanced nodes and can be reconstructedin an incremental manner without retraining the entire network fromscratch. On the premise of ensuring that the accuracy meets therequirements, the present invention continuously and rapidly adjustsitself as the production process changes, realizes timely online faultsmonitoring of sewage treatment process, and obtains good monitoringresults.

DESCRIPTION OF THE DRAWINGS

FIG. 1 is a diagram of the overall structure of the present invention;

FIG. 2 is a structural diagram of i-th fuzzy subsystem of the presentinvention;

FIG. 3 shows the monitoring result of fault 1;

FIG. 4 shows the monitoring results of fault 2;

FIG. 5 shows the monitoring results of fault 3;

FIG. 6 shows the monitoring results of fault 4;

FIG. 7 shows the logic block diagram of the hardware system on whichthis method is based.

PREFERRED EMBODIMENT

In order to solve the above-mentioned problems, a fault monitoringmethod of sewage treatment process based on fuzzy width adaptivelearning model is proposed, which is based on an online monitoringequipment. The whole device includes input module, informationprocessing module, console module, output result visualization module.The proposed method is imported into the information processing module,and then the network monitoring model is established with the processdata retained by the actual industry, and the established model is savedfor online fault monitoring. In actual online monitoring of industrialprocesses, first connect real-time process variables collected by thefactory data sensor to input module as the input information ofmonitoring equipment, and then select previously trained model throughconsole to monitor, and the monitoring results is displayed in real timethrough visualization module, so that on-site staff can make timelycorresponding measures based on visualization monitoring results toreduce the economic loss caused by process failures.

The sewage treatment process is extremely complex, including not only avariety of physics and chemistry, but also biochemical reactions. Inaddition, various uncertain factors are flooded with it, such asinfluent flow, water quality and load changes, which give hugechallenges for establishment of sewage treatment monitoring model. Thepresent invention uses the “benchmark simulation model 1” (benchmarksimulation model 1) developed by the International Water Association(IWA) as the actual sewage treatment process for real-time simulation.The model consists of five reaction tanks (5999 m³) and a secondarysettling tank (6000 m³), in addition, it has three aeration tanks. Theaeration tank has 10 layers, 4 meters deep, and covers an area of 1500m². The reaction process includes internal and external reflux. Theaverage sewage treatment flow rate is 20 000 m³/d, and the chemicaloxygen demand is 300 mg/l.

This experiment takes BSM1 sewage data as the research object, thesampling interval is 15 minutes, and the sampling period is 14 days.That is, there are 96 sampling points every day, totaling 1344 samplingpoints. Select 16 variables to monitor the operating status of theprocess, as shown in Table 1. A batch of normal data and 4 batches offault data are used as training data to train the network offline, and 4batches of fault data are used as test data to verify the effectivenessof the method. Set the fault types of two types of faults under rainyconditions: single fault and compound fault. The fault occurrence timeis both the eighth day and the ninth day. The detailed information ofthe fault is shown in Table 2.

TABLE 1 Sewage variables Number Name of Variables Unit 1 Discharge fluxm−3 · d 2 Discharge SI g COD · m⁻³ concentration 3 Discharge SS g COD ·m⁻³ concentration 4 Discharge XI g COD · m⁻³ concentration 5 DischargeXS g COD · m⁻³ concentration 6 Discharge XBH g COD · m⁻³ concentration 7Discharge XBA g COD · m⁻³ concentration 8 Discharge XP g COD · m⁻³concentration 9 Discharge SO g (−COD) · m⁻³ concentration 10 DischargeSNO g N · m⁻³ concentration 11 Discharge SNH g N · m⁻³ concentration 12Discharge SND g N · m⁻³ concentration 13 Discharge XND g N · m⁻³concentration 14 Discharge SALK mol HCO3− · m⁻³ concentration 15Discharge TSS g SS · m⁻³ concentration 16 Discharge Kjeldahl N g N · m⁻³concentration

TABLE 2 Failure data settings Failure Working Failure Starting Endingcode condition Amplitude type time time 1 Rainy Muh = 3.5 Single 673 8642 Rainy Muh = 1 Single 673 864 3 Rainy Muh = 2 BH = 0.45 Compound 673864 4 Rainy Muh = 2 BH = 0.6 Compound 673 864

The fault monitoring process of the application of the present inventionin the above-mentioned BSM1 sewage treatment process is specificallystated as follows:

A. Network Training Stage:

Step 1: Collect a batch of operating data under normal workingconditions and 4 batches of fault data as a training data X∈

^(N×M) set through BSM1 platform. Each batch of data samples has 1344points, 16 variables. Take five sets of data as a training data set. Itcan be expressed as X (6720×16), label the data, the label value is 0 atnormal time, and the label value at fault time is 1, which isrepresented as Y (6720×2).

Step 2: The number of fuzzy subsystems, enhanced node groups, and fuzzyrules of each fuzzy subsystem are determined according to the actualsituation. The model in this embodiment contains n=8 fuzzy subsystems,m=10 enhanced node groups, and each group contains 10 neurons. Thenumber of fuzzy rules for each fuzzy subsystem is 5, and the first-orderT-S is adopted. Fuzzy model, taking the i-th fuzzy subsystem as anexample (i=1, 2, . . . , n), when the input data is s-th trainingsample, the output of k-th fuzzy rule of i-th fuzzy subsystem is:

$z_{sk}^{i} = {{f_{k}^{i}\left( {x_{s\; 1},x_{s\; 2},\ldots\;,x_{sM}} \right)} = {\sum\limits_{t = 1}^{M}\;{\alpha_{kt}^{i}x_{st}}}}$

Wherein, f_(k) ^(i) is a polynomial of the first degree, α_(kt) ^(i) isa parameter, obeys uniform distribution [0,1], t=1, 2, . . . , M, k=1,2, . . . , K represents the number of variables.

Step 3: For the s-th training sample, calculate weighted strength ω_(sk)^(i) of k-th fuzzy rule of the i-th fuzzy subsystem:

$\omega_{sk}^{i} = \frac{\tau_{sk}^{i}}{\sum\limits_{k = 1}^{k_{i}}\;\tau_{sk}^{i}}$

Where τ_(sk) ^(i) represents weighted strength of the kth fuzzy rule ofi-th fuzzy subsystem (k=1, 2, . . . , K) when the input training sampleis Xs (xs=(xs1, xs2, . . . , xsM)), the formula is as follows:

$\tau_{sk}^{i} = {\prod\limits_{t = 1}^{M}\;{\mu_{kt}^{i}\left( x_{st} \right)}}$

Where μ_(kt) ^(i)(x_(st)) is the Gaussian membership function (t=1, 2, .. . , M) corresponding to the input sample X_(st), and its formula isexpressed as:

${\mu_{st}^{i}\left( x_{st} \right)} = e^{- {(\frac{x_{st} - c_{kt}^{i}}{\delta_{kt}^{i}})}^{2}}$

Wherein c_(kt) ^(i) and δ_(kt) ^(i) are parameters, and the value ofc_(kt) ^(i) is the value of cluster center obtained by C-meansclustering, δ_(kt) ^(i)=1

Step 4: Calculate the output Z^(n) of n fuzzy subsystems correspondingto sth training sample:Z ^(n)=(Z ₁ ,Z ₂ , . . . ,Z _(n))∈

^(N×(k) ¹ ^(+k) ² ^(+ . . . k) ^(n) ⁾

Wherein, Z₁, Z₂, . . . , Z₁ respectively correspond to the output of nsubsystems, k₁, k₂, . . . , k_(n) respectively correspond to the numberof fuzzy rules of n fuzzy subsystems, and the sth training sample.

Taking i-th fuzzy subsystem as an example, for all training samples, theoutput matrix Z_(i) is obtained by the following formula:Z _(i)=(Z _(1i) ,Z _(2i) , . . . ,Z _(Ni))^(T)∈

^(N×K) ,i=1,2, . . . ,n

Wherein, Z_(1i), Z_(2i), . . . , Z_(Ni) respectively represent theoutput vector of i-th fuzzy subsystem for N training samples. Takings-th sample as an example, the output vector of i-th fuzzy subsystem canbe obtained by the following formula:Z _(si)=(ω_(s1) ^(i) z _(s1) ^(i),ω_(s2) ^(i) z _(s2) ^(i), . . .,ω_(sk) ₁ ^(i) z _(sk) ₁ ^(i))

Wherein, z_(sk) ^(i) and ω_(sk) ^(i) are obtained by 2) and 3)respectively, and k=1, 2, . . . , K.

Step 5: Input the output Z^(n) obtained in step 4 into m enhanced nodegroups mentioned in step 2 for nonlinear conversion. Wherein, the j-thenhanced node group has Lj neurons (=1, 2, . . . , m). For m enhancednode groups, the output matrix can be expressed as:H ^(m)=(H ₁ ,H ₂ , . . . ,H _(m))∈

^(N×(L) ¹ ^(+L) ² ^(+ . . . L) ^(m) ⁾

Wherein, L₁, L₂, . . . , L_(m) respectively represent the number ofneurons contained in m enhanced node groups,H_(j)=ξ_(j)(Z^(n)W_(j)+β_(j))∈

^(N×L) ^(j) is the output matrix of j-th enhanced node group, and Wjrepresents connection weight between fuzzy subsystem output and thecorresponding enhanced node group, βj represents the deviation betweenoutput of fuzzy subsystem and j-th enhanced node group, the values of Wjand βj are randomly generated from [0,1], ξj is the activation function,and the formula is ξj=tan h(.).

Step 6: Calculate the defuzzification output of fuzzy subsystem. Thenfor s-th training sample X_(s), the defuzzification output vector ofi-th fuzzy subsystem can be expressed as:

$F_{si} = {{\sum\limits_{k = 1}^{k_{i}}\;{\omega_{sk}^{i}z_{sk}^{i}}} = {\sum\limits_{k = 1}^{k_{i}}\;\left( {\omega_{sk}^{i}{\sum\limits_{t = 1}^{M}\;{\alpha_{kt}^{i}{xs}_{st}}}} \right)}}$

Introduce parameter δ_(kc) ^(i) to the result part of each fuzzy rule.The parameter value is obtained by pseudo-inverse operation method, andC represents dimension of output. Since Y=∈

^(N×C) has C dimensions, each fuzzy subsystem is a multi-output model.After the parameter is introduced, its new output is:

$\begin{matrix}{F_{si} = \left( {{\sum\limits_{k = 1}^{k_{i}}\;{\omega_{sk}^{i}\left( {\sum\limits_{t = 1}^{M}\;{\delta_{k\; 1}^{i}\alpha_{kt}^{i}x_{st}}} \right)}},\ldots\;,{\sum\limits_{k = 1}^{K}\;{\omega_{sk}^{i}\left( {\sum\limits_{t = 1}^{M}\;{\delta_{kC}^{i}\alpha_{kt}^{i}x_{st}}} \right)}}} \right)} \\{= {\sum\limits_{t = 1}^{M}\;{\alpha_{kt}^{i}{x_{st}\left( {\omega_{s\; 1}^{i},\omega_{s\; 2}^{i},\ldots\;,\omega_{sK}^{i}} \right)}\begin{pmatrix}\delta_{11}^{i} & \cdots & \delta_{1C}^{i} \\\vdots & \ddots & \vdots \\\delta_{K\; 1}^{i} & \cdots & \delta_{KC}^{i}\end{pmatrix}}}}\end{matrix}$

Where s=1, 2, . . . , N, then, for all training samples X, the output ofthe i-th fuzzy subsystem is:

$F_{i} = {\left( {F_{1i},F_{2i},\ldots\;,F_{Ni}} \right)^{T}\overset{\Delta}{=}{{D\;\Omega^{i}\delta^{i}} \in {\mathbb{R}}^{N \times C}}}$${{{Where}\mspace{14mu} D} = {{diag}\left\{ {{\sum\limits_{t = 1}^{M}\;{\alpha_{kt}^{i}x_{1t}}},\ldots\;,{\sum\limits_{t = 1}^{M}\;{\alpha_{kt}^{i}x_{Nt}}}} \right\}}},{\Omega^{i} = \begin{pmatrix}\omega_{11}^{i} & \cdots & \omega_{1K}^{i} \\\vdots & \ddots & \vdots \\\omega_{N\; 1}^{i} & \cdots & \omega_{NK}^{i}\end{pmatrix}},{\delta^{i} = \begin{pmatrix}\delta_{11}^{i} & \cdots & \delta_{1C}^{i} \\\vdots & \ddots & \vdots \\\delta_{K\; 1}^{i} & \cdots & \delta_{KC}^{i}\end{pmatrix}}$

Thus, the output of n fuzzy subsystems can be obtained:

$F^{n} = {{\sum\limits_{i = 1}^{n}\; F_{i}} = {{\sum\limits_{i = 1}^{n}\;{D\;\Omega^{i}\delta^{i}}} = {{{D\left( {\Omega^{1},\Omega^{2},\ldots\;,\Omega^{n}} \right)}\begin{pmatrix}\delta^{1} \\\vdots \\\delta^{n}\end{pmatrix}} = {{D\;{\Omega\Delta}} \in {\mathbb{R}}^{N \times C}}}}}$

Wherein, Ω=(Ω¹, Ω², . . . , Ω^(n))∈

^(N×nk) is the matrix containing the weighted strengthen ω_(sk) ^(i)obtained in 3). Δ=((δ¹)^(T), . . . , (δ^(n))^(T))^(T)∈

^(nK×C)

Step 7: Calculate the final output result. Weight between fuzzysubsystem and output is set to 1, and the weight between enhanced nodegroup and output is set to W_(e)∈

^((L) ¹ ^(+L) ² ^(+ . . . L) ^(m) ^()×C). Multiply the H^(m) obtained in4) and F^(n) obtained in 5) respectively. The weights are then summed toget the final output result, the formula is as follows:

$\hat{Y} = {{F^{n} + {H^{m}W_{e}}} = {{{D\;{\Omega\Delta}} + {H^{m}W_{e}}} = {{\left( {{D\;\Omega},H^{m}} \right)\begin{pmatrix}\Delta \\W_{e}\end{pmatrix}}\overset{\Delta}{=}{\left( {{D\;\Omega},H^{m}} \right)W}}}}$

For a given training target Y, the matrix W can be calculated bypseudo-inverse:W=(DΩ,H ^(m))⁺ Y

Wherein, so far, the network model training is complete.

B Online Monitoring:

Step 8: During online monitoring, set the new data X_(new)=(x_(new,1),x_(new,2), . . . , x_(new,N))^(T)∈

^(N×M), X_(new,s)=(X_(new,1), X_(new,2), . . . , X_(new,M)), s=1, 2, . .. , N. For n fuzzy subsystems in the training part, taking i-thsubsystem as an example, using first-order T-S fuzzy system, then:

$z_{{new},{sk}}^{i} = {{f_{k}^{i}\left( {x_{{new},{s\; 1}},x_{{new},{s\; 2}},\ldots\;,x_{{new},{sM}}} \right)} = {\sum\limits_{t = 1}^{M}\;{\alpha_{kt}^{i}x_{{new},{st}}}}}$

The value of α_(kt) ^(i) is the value corresponding to the trainingpart.

Step 9: For s-th test sample, the weighted strength of i-th fuzzysubsystem is as follows:

$\omega_{{new},{sk}}^{i} = \frac{\tau_{{new},{sk}}^{i}}{\sum\limits_{k = 1}^{K}\;\tau_{{new},{sk}}^{i}}$

Where τ_(new,sk) ^(i) represents the strength of k-th fuzzy rule of i-thfuzzy subsystem, which is obtained by the following formula:

$\tau_{{new},{sk}}^{i} = {\prod\limits_{t = 1}^{M}\;{\mu_{kt}^{i}\left( x_{{new},{st}} \right)}}$

Where μ_(kt) ^(i)(x_(new,st)) is the Gaussian membership functioncorresponding to the input sample X_(new,st) (t=1, 2, . . . , M), andits formula is expressed as:

${\mu_{kt}^{i}\left( x_{{new},{st}} \right)} = e^{- {(\frac{x_{{new},{st}} - c_{kt}^{i}}{\delta_{kt}^{i}})}^{2}}$

Wherein, c_(kt) ^(i) and δ_(kt) ^(i) are parameters, and their valuesare the same as corresponding parameter values of the training part.

Step 10: For all test samples, find the output Z_(new) ^(n) of n fuzzysubsystems. The formula is as follows:Z _(new) ^(n)=(Z _(new1) ,Z _(new2) , . . . ,Z _(newn))∈

^(N×nK)

The output Z_(newi) of i-th fuzzy subsystem can be expressed as follows:Z _(newi)=(Z _(new,1i) ,Z _(new,2i) , . . . ,Z _(new,Ni))^(T)∈

^(N×K) ,i=1,2, . . . ,n

Taking s-th training sample among all N samples as an example, theoutput vector Z_(new,si) of i-th fuzzy subsystem can be expressed as:Z _(new,si)=(ω_(new,s1) ^(i) z _(new,s1) ^(i),ω_(new,s2) ^(i) z_(new,s2) ^(i), . . . ,ω_(new,sK) ^(i) z _(newsK) ^(i))

Where (s=1, 2, . . . , N)

Wherein, Z_(new,sk) ^(i) and ω_(new,sk) ^(i) are obtained by 8) and 9)respectively, k=1, 2, . . . , K.

Step 11: The output Z_(new) ^(n) obtained in 10) is subjected tononlinear conversion.

Then for m enhanced node groups, the output matrix of enhanced layer canbe expressed as:H _(new) ^(m)=(H _(new1) ,H _(new2) , . . . ,H _(newm))∈

^(N×(L) ¹ ^(+L) ² ^(+ . . . L) ^(m) ⁾

Where H_(newj)=ξ_(j)(Z_(new) ^(n)W_(j)+β_(j))∈

^(N×L) ^(j) is the output matrix of j-th enhanced node group (j=1, 2, .. . , m), and the values of Wj and βj correspond to the network trainingpart.

Step 12: Calculate the defuzzification output of fuzzy subsystem.

After introducing parameter δ_(kc) ^(i) to the result part of each fuzzyrule, c=1, 2, . . . , C. For s-th training sample, defuzzificationoutput vector of i-th fuzzy subsystem can be expressed as:

$\begin{matrix}{F_{newsi} = \left( {\sum\limits_{k = 1}^{K_{i}}\;{{\omega_{sk}^{i}\left( {\sum\limits_{t = 1}^{M}\;{\delta_{k\; 1}^{i}\alpha_{kt}^{i}x_{newst}}} \right)}\mspace{14mu}\ldots\mspace{14mu}{\sum\limits_{k = 1}^{K_{i}}\;{\omega_{sk}^{i}\left( {\sum\limits_{t = 1}^{M}\;{\delta_{kC}^{i}\alpha_{kt}^{i}x_{newst}}} \right)}}}} \right)} \\{= {\sum\limits_{t = 1}^{M}\;{\alpha_{kt}^{i}{x_{newst}\left( {\omega_{s\; 1}^{i},\omega_{s\; 2}^{i},\ldots\;,\omega_{sK}^{i}} \right)}\begin{pmatrix}\delta_{11}^{i} & \cdots & \delta_{1C}^{i} \\\vdots & \ddots & \vdots \\\delta_{K1}^{i} & \cdots & \delta_{KC}^{i}\end{pmatrix}}}}\end{matrix}$

Then, for all training samples X, the output of i-th fuzzy subsystem is:

$F_{newi} = {\left( {F_{{new},{1i}},F_{{new},{2i}},\ldots\;,F_{{new},{Ni}}} \right)^{T}\overset{\Delta}{=}{{D_{new}\Omega_{new}^{i}\delta^{i}} \in {\mathbb{R}}^{N \times C}}}$${{{Wherein}\mspace{14mu} D_{new}} = {{diag}\left\{ {{\sum\limits_{t = 1}^{M}\;{\alpha_{kt}^{i}x_{{new},{1t}}}},\ldots\;,{\sum\limits_{t = 1}^{M}\;{\alpha_{kt}^{i}x_{{new},{Nt}}}}} \right\}}},{\Omega_{new}^{i} = \begin{pmatrix}\omega_{{new},11}^{i} & \cdots & \omega_{{{new}\; 1},K}^{i} \\\vdots & \ddots & \vdots \\\omega_{{new},{N\; 1}}^{i} & \cdots & \omega_{{new},{NK}}^{i}\end{pmatrix}},{\delta^{i} = \begin{pmatrix}\delta_{11}^{i} & \cdots & \delta_{1C}^{i} \\\vdots & \ddots & \vdots \\\delta_{K\; 1}^{i} & \cdots & \delta_{KC}^{i}\end{pmatrix}}$

Thus, the output of n fuzzy subsystems can be obtained:

$F_{new}^{n} = {{\sum\limits_{i = 1}^{n}\; F_{newi}} = {{\sum\limits_{i = 1}^{n}\;{D_{new}\Omega_{new}^{i}\delta^{i}}} = {{{D_{new}\left( {\Omega_{new}^{1},\ldots,\Omega_{new}^{n}} \right)}\begin{pmatrix}\delta^{1} \\\vdots \\\delta^{n}\end{pmatrix}} = {{D_{new}\Omega_{new}\Delta} \in {\mathbb{R}}^{N \times C}}}}}$

Wherein, Ω_(new)=(Ω_(new) ¹, . . . , Ω_(new) ^(n))∈

^(N×nK) is the matrix containing weighted strength ω_(new,sk) ^(i)obtained in 9).Δ=((δ¹)^(T), . . . ,(δ^(n))^(T))^(T)∈

^(nK×C)

Step 13: Calculate the final output value. Set the weight betweencombined output of connection enhancement node to W_(e)∈

^((L) ¹ ^(+L) ² ^(+ . . . L) ^(m) ^()×C) whose value is obtained fromtraining phase, and the weight between fuzzy subsystem and output is 1,multiply H_(new) ^(m) obtained from step 11 and F_(new) ^(n) obtainedfrom step 12 by their respective weights, and then sum them to get finaloutput. The formula is as follows:Ŷ=F _(new) ^(n) +H _(new) ^(m) W _(e)

(D _(new)Ω_(new) ,H _(new) ^(m))W

The value of W corresponds to the value of parameter matrix of networktraining part.(D _(new)Ω_(new) ,H _(new) ^(m))₊=((D _(new)Ω_(new) ,H _(new)^(m))^(T)(D _(new)Ω_(new) ,H _(new) ^(m)))⁻¹(D _(new)Ω_(new) ,H _(new)^(m))^(T)

Step 14: The output value Ŷ is converted to zero by one-hot encodingmethod. When it is 0, it means that the operation is in a normal state.When the output value is 1, it means that the operation is faulty, so asto realize fault monitoring.

The above steps are specific application of the method of the presentinvention in fault monitoring of the sewage treatment process of BSM1platform. In the experiment, two types of faults under sunny and dryweather conditions are used, namely single fault and compound fault, andthe fault occurrence time is both on the eighth and ninth days. Theexperimental results are shown in FIG. 3 to FIG. 6. Abscissa of eachpicture represents the sampling point, and the ordinate is the outputresult, showing a judgment of 0, 1, where 0 means normal operation and 1means malfunction. The performance indicators are false alarm rate(FAR), missed alarm rate (MAR) and accuracy rate (ACC), among which:FAR=The number of false alarm/total normal time*100%MAR=The number of missed alarm/total failure time*100%ACC=(The number of false alarm+The number of missed alarm)/totalsampling time*100%

Unilateral consideration of false alarm rate (FAR) or missed alarm rate(MAR) cannot fully reflect the quality of monitoring performance.Therefore, the false alarm rate (FAR) or missed alarm rate (MAR) shallbe considered comprehensively. When the false alarm rate (When FAR andMAR are both low, that is, when the accuracy rate (ACC) is low, themonitoring performance is considered better.

Aiming at the situation of more rainy weather in summer in North China,this method specifically conducts monitoring experiments under rainyconditions. The experimental faults are divided into single faults andcompound faults. For a single type of failure, the fuzzy rule for eachsubsystem is 8, and the enhanced node group is 10. For fault 1, thefault amplitude muh=3.5, the start time is 673, and the end time is 864.The monitoring results are shown in FIG. 3. There are no false positivesand false negatives. It has good monitoring performance, and the networktraining time is 4.85 s, which can detect the occurrence of faults intime. Under the same fault type, change the magnitude of the fault sothat when muh=1, the fault occurrence time is the same. The monitoringresults are shown in FIG. 4. The network training time is 4.67 s, andthe occurrence of the fault can be monitored immediately at thebeginning of the fault which has good monitoring performance.

Subsequently, the fault type under rainy conditions was changed to acomposite fault for verification. The system has a total of 10 fuzzysubsystems, each of which contains 8 fuzzy rules, and there are 10enhanced node groups. For fault 3, the amplitude is muh=2, BH=0.45, andthe false alarm rate is 0. At 720, there are 5 false alarms, and thefalse alarm rate is 2.62%. Considering the false alarm and the falsealarm, the accuracy rate is ACC. It is 0.37%, which has a goodmonitoring effect overall, and the time used is 7.85 s, which can detectthe occurrence of the fault in time. For fault 4, changing the faultamplitude to muh=2 and BH=0.45, the monitoring effect has not changed,and the detection performance can also be exerted well, and the networktraining time is 8.07 s. Judging from the monitoring effect of the above4 failures, compared with the deep neural network, the overallperformance of the method of the present invention is lesstime-consuming for network training under the condition that themonitoring effect is satisfactory, and the sewage treatment is greatlyreduced. With the consumption of hardware resources in the plant,ordinary computers can do the job, which saves costs and enables timelyonline monitoring of faults in the sewage treatment process.

What is claimed is:
 1. A fault monitoring method for sewage treatmentprocess based on fuzzy width adaptive learning model comprising: anetwork training phase and an online monitoring phase; the specificsteps are as follows: A) the network training stage: 1) collecthistorical data {X, Y} of the sewage treatment process as training data,where X=(x₁, x₂, . . . , x_(N))^(T)∈

^(N×M), Y=(y₁, y₂, . . . , y_(C))^(T)∈

^(N×C), Y is used as label data, the label data is 0 during a normalperiod of the sewage treatment process, and the label data is 1 during afault period; N is the number of input samples, M is the number ofvariables, and C is an output dimension, wherein the output dimension is2, which means normal and faulty states; 2) for input data X=(x₁, x₂, .. . , x_(N))^(T)∈

^(N×M), X_(s)=(x_(s1), xs2, . . . , x_(sM)), s=1, 2, . . . , N; themodel has n fuzzy subsystems, m enhanced node groups, and i-th fuzzysubsystem contains K fuzzy rules; a first-order TS fuzzy model is used,where the i-th fuzzy subsystem is taken as an example (i=1, 2, . . .,n); when the input data is s-th training sample, an output of i-thfuzzy subsystem and k-th fuzzy rule is:$z_{sk}^{i} = {{f_{k}^{i}\left( {x_{s\; 1},x_{s2},\ldots\mspace{14mu},x_{s\; M}} \right)} = {\sum\limits_{t = 1}^{M}{a_{kt}^{i}x_{st}}}}$wherein f_(i) ^(k) is a polynomial of a first degree, α_(kt) ^(i) is aparameter obtained by initialization which obeys uniform distribution[0,1], t=1, 2, . . . ,M, k=1, 2, . . . , K is the number of variables;3) for the s-th training sample, calculate weighted strength ω_(sk) ^(i)of k-th fuzzy rule of the i-th fuzzy subsystem:$\omega_{sk}^{i} = \frac{\tau_{sk}^{i}}{\sum\limits_{k = 1}^{k_{i}}\tau_{sk}^{i}}$where τ_(sk) ^(i) represents weighted strength of kth fuzzy rule of i-thfuzzy subsystem (k=1, 2, . . . ,K) when an input training sample is Xs(xs=(xs1, xs2, . . . ,xsM)), the formula is as follows:$\tau_{sk}^{i} = {\prod\limits_{t = 1}^{M}\;{\mu_{kt}^{i}\left( x_{st} \right)}}$where μ_(kt) ^(i)(x_(st)) is Gaussian membership function (t=1, 2, . . ., M) corresponding to the input sample X_(st), and its formula isexpressed as:${\mu_{kt}^{i}\left( x_{st} \right)} = e^{- {(\frac{x_{st} - c_{kt}^{i}}{\delta_{kt}^{i}})}^{2}}$c_(kt) ^(i) and δ_(kt) ^(i) are parameters, and the value of c_(kt) ^(i)is the value of cluster center obtained by C-means clustering, δ_(kt)^(i)=1; 4) Calculate the output Z^(n) of n fuzzy subsystemscorresponding to sth training sample:Z ^(n)=(Z ₁ ,Z ₂ , . . . ,Z _(n))∈

^(N×(k) ¹ ^(+k) ² ^(+ . . . k) ^(n) ⁾ Wherein, Z₁, Z₂, . . . , Z_(n)respectively correspond to the output of n subsystems, k₁, k₂, . . . ,k_(n) respectively correspond to the number of fuzzy rules of n fuzzysubsystems, and the sth training sample; taking i-th fuzzy subsystem asan example, for all training samples, an output matrix Z_(i) is obtainedby the following formula:Z _(i)=(Z _(1i) ,Z _(2i) , . . . ,Z _(Ni))^(T)∈

^(N×K) ,i=1,2, . . . ,n Wherein, Z_(1i), Z_(2i), . . . , Z_(Ni)respectively represent an output vector of i-th fuzzy subsystem for Ntraining samples; taking s-th sample as an example, the output vector ofi-th fuzzy subsystem can be obtained by the following formula:Z _(si)=(ω_(s1) ^(i) z _(s1) ^(i),ω_(s2) ^(i) z _(s2) ^(i), . . .,ω_(sk) ₁ ^(i) z _(sk) ₁ ^(i)) Wherein, z_(sk) ^(i) and ω_(sk) ^(i) andare obtained by 2) and 3) respectively, and k=1, 2, . . . ,K; 5) inputthe output Z^(n) obtained in 4) into m enhanced node groups mentioned in2) for nonlinear conversion; wherein, j-th enhanced node group has Ljneurons (j=1, 2, . . . ,m); for m enhanced node groups, an output matrixcan be expressed as:H ^(m)=(H ₁ ,H ₂ , . . . ,H _(m))∈

^(N×(L) ¹ ^(+L) ² ^(+ . . . L) ^(m) ⁾ wherein, L₁, L₂, . . . , L_(m)respectively represent the number of neurons contained in m enhancednode groups, H₁=ξ_(j)(Z^(n)W_(j)+β_(j))∈

^(N×L) ^(j) is an output matrix of j-th enhanced node group, and Wjrepresents connection weight between fuzzy subsystem output and thecorresponding enhanced node group, βj represents the deviation betweenoutput of fuzzy subsystem and j-th enhanced node group, the values of Wjand βj are randomly generated from [0,1], ξj is the activation function,and the formula is ξj=tan h(·); 6) calculate defuzzification output offuzzy subsystem to obtain D, Ω, for s-th training sample X_(s), adefuzzification output vector of i-th fuzzy subsystem can be expressedas: $\begin{matrix}{F_{si} = \left( {{\sum\limits_{k = 1}^{k_{i}}\;{\omega_{sk}^{i}\left( {\sum\limits_{t = 1}^{M}\;{\delta_{k\; 1}^{i}\alpha_{kt}^{i}x_{st}}} \right)}},\ldots\;,{\sum\limits_{k = 1}^{K}\;{\omega_{sk}^{i}\left( {\sum\limits_{t = 1}^{M}\;{\delta_{kC}^{i}\alpha_{kt}^{i}x_{st}}} \right)}}} \right)} \\{= {\sum\limits_{t = 1}^{M}\;{\alpha_{kt}^{i}{x_{st}\left( {\omega_{s\; 1}^{i},\omega_{s\; 2}^{i},\ldots\;,\omega_{sK}^{i}} \right)}\begin{pmatrix}\delta_{11}^{i} & \cdots & \delta_{1C}^{i} \\\vdots & \ddots & \vdots \\\delta_{K\; 1}^{i} & \cdots & \delta_{KC}^{i}\end{pmatrix}}}}\end{matrix}$ where s=1,2, . . . ,N, then, for all training samples X,the output of the i-th fuzzy subsystem is:$F_{i} = {\left( {F_{1i},F_{2i},\ldots\;,F_{Ni}} \right)^{T}\overset{\Delta}{=}{{D\;\Omega^{i}\delta^{i}} \in {\mathbb{R}}^{N \times C}}}$${{{where}\mspace{14mu} D} = {{diag}\left\{ {{\sum\limits_{t = 1}^{M}\;{\alpha_{kt}^{i}x_{1t}}},\ldots\;,{\sum\limits_{t = 1}^{M}\;{\alpha_{kt}^{i}x_{Nt}}}} \right\}}},{\Omega^{i} = \begin{pmatrix}\omega_{11}^{i} & \cdots & \omega_{1K}^{i} \\\vdots & \ddots & \vdots \\\omega_{N\; 1}^{i} & \cdots & \omega_{NK}^{i}\end{pmatrix}},{\delta^{i} = \begin{pmatrix}\delta_{11}^{i} & \cdots & \delta_{1C}^{i} \\\vdots & \ddots & \vdots \\\delta_{K\; 1}^{i} & \cdots & \delta_{KC}^{i}\end{pmatrix}}$ thus, the output of n fuzzy subsystems can be obtained:${F^{n} - {\sum\limits_{i = 1}^{n}\; F_{i}} - {\sum\limits_{i = 1}^{n}\;{D\;\Omega^{i}\delta^{i}}}} = {{{{D\left( {\Omega^{1},\Omega^{2},\ldots\;,\Omega^{n}} \right)}\begin{pmatrix}\delta^{1} \\\vdots \\\delta^{n}\end{pmatrix}} - {D\;{\Omega\Delta}}} \in {\mathbb{R}}^{N \times C}}$wherein, Ω=(Ω¹, Ω², . . . , Ω^(n))∈

^(N×nk) is the matrix containing the weighted strengthen ω_(sk) ^(i)obtained in 3); Δ=((δ¹)^(T), . . . , (δ^(n))^(T))^(T)∈

^(nK×C) 7) calculate network parameter W, the formula is as follows:$\hat{Y} = {{F^{n} + {H^{m}W_{e}}} = {{{D\;{\Omega\Delta}} + {H^{m}W_{e}}} = {{\left( {{D\;\Omega},H^{m}} \right)\begin{pmatrix}\Delta \\W_{e}\end{pmatrix}}\overset{\Delta}{=}{\left( {{D\;\Omega},H^{m}} \right)W}}}}$wherein Ŷ represents a predicted output of the network model D, Ω, H^(m)can be obtained by previous steps, by inputting training data Y, theparameter W in the network model can be obtained; the specific formulais as follows:W=(DΩ,H ^(m))⁺ Y wherein, (DΩ, H^(m))⁺=((DΩ,H^(m)))^(T) (DΩ,H^(m)))⁻¹(DΩ,H^(m))^(T), so far, the network model training is complete;B) the online monitoring phase: based on the X_(new) obtained fromreal-time monitoring, new D, Ω, a H^(m) are calculated; wherein,parameters involved in calculation process are consistent with thenetwork training stage, and then the network model completed by trainingis used for prediction, and an output value Ŷ is converted to zero orone by means of one-hot encoding, wherein zero indicates that theoperation is in a normal state; when the output value is temporary, itindicates that the operation is in fault, so as to realize faultmonitoring.